Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2007 Polish Mathematical Olympiad Third Round"

(Created page with "==Day 1== ===Problem 1=== In an acute triangle <math>ABC</math> let <math>O</math> be the center of the circumcircle, segment <math>CD</math> be the height, point <math>E</mat...")
 
(→‎Problem 5)
 
Line 14: Line 14:
  
 
===Problem 5===
 
===Problem 5===
−
Tetrahedron <math>ABCD</math> satisfies the equalities <cmath>\angle BAC + \angle BDC = \angle ABD + \angle ACD,\\ \angle BAD + \angle BCD = \angle ABC + \angle ADC.</cmath> Prove that the center of sphere circumscribed on <math>ABCD</math> lies on the line passing through the centres of edges <math>AB</math> and <math>CD</math>.
+
Tetrahedron <math>ABCD</math> satisfies the equalities <cmath>\angle BAC + \angle BDC = \angle ABD + \angle ACD,</cmath> <cmath>\angle BAD + \angle BCD = \angle ABC + \angle ADC.</cmath> Prove that the center of sphere circumscribed on <math>ABCD</math> lies on the line passing through the centres of edges <math>AB</math> and <math>CD</math>.
  
 
===Problem 6===
 
===Problem 6===
 
Sequence <math>a_0, a_1, a_2, \ldots</math> is described by conditions: <math>a_0 = -1</math> and <cmath>a_n + \frac{a_{n-1}}{2} + \frac{a_{n-2}}{3} + \ldots + \frac{a_1}{n} + \frac{a_0}{n+1}=0 \qquad \text{for } n \geq 1.</cmath>
 
Sequence <math>a_0, a_1, a_2, \ldots</math> is described by conditions: <math>a_0 = -1</math> and <cmath>a_n + \frac{a_{n-1}}{2} + \frac{a_{n-2}}{3} + \ldots + \frac{a_1}{n} + \frac{a_0}{n+1}=0 \qquad \text{for } n \geq 1.</cmath>
 
Prove that <math>a_n > 0</math> for <math>n \geq 1</math>.
 
Prove that <math>a_n > 0</math> for <math>n \geq 1</math>.

Latest revision as of 15:08, 4 July 2022

Day 1

Problem 1

In an acute triangle $ABC$ let $O$ be the center of the circumcircle, segment $CD$ be the height, point $E$ be any point on the segment $AB$ and point $M$ be the middle of segment $CE$. The line perpendicular to $OM$ and passing through $M$ passes the lines $AC, BC$ accordingly in points $K,L$. Prove that \[\frac{LM}{MK}=\frac{AD}{DB}\].

Problem 2

We call a positive integer $\textit{white}$ if it equals 1 or is the product of an even number of prime numbers (not necessarily distinct). All the other positive integers will be called $\textit{black}$. Examine if there exists any positive integer such that sum of its white divisors equals the sum of its black divisors.

Problem 3

The plane has been divided with horizontal and vertical lines into unit squares. In each square we write one positive integer so that every positive integer appears only once in the plane. Is it possible to write the numbers in such a way that every number is the divisor of the sum of numbers in 4 neighbour squares.?

Day 2

Problem 4

Let $n \geq 1$ be an integer. Determine the number of possible values of the product $km$, where $k,m$ are integers satisfying the inequalities \[n^2 \leq k \leq m \leq (n+1)^2\].

Problem 5

Tetrahedron $ABCD$ satisfies the equalities \[\angle BAC + \angle BDC = \angle ABD + \angle ACD,\] \[\angle BAD + \angle BCD = \angle ABC + \angle ADC.\] Prove that the center of sphere circumscribed on $ABCD$ lies on the line passing through the centres of edges $AB$ and $CD$.

Problem 6

Sequence $a_0, a_1, a_2, \ldots$ is described by conditions: $a_0 = -1$ and \[a_n + \frac{a_{n-1}}{2} + \frac{a_{n-2}}{3} + \ldots + \frac{a_1}{n} + \frac{a_0}{n+1}=0 \qquad \text{for } n \geq 1.\] Prove that $a_n > 0$ for $n \geq 1$.

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2007_Polish_Mathematical_Olympiad_Third_Round&oldid=175586"